Guys,
My barrels almost never are on center either. Unless a miracle happens! But, even a blind hog finds an acorn every once in a while. I file the barrel outside to bring the bore on center. Very simple, actually. Just a little time consuming. You can also check the barrel wall thickness dimension with a really simple tool. I will try to describe it in words as I do not have access to photos here.
Take a 2 foot round rod, 3/8 inch diameter. Thread one end for a jag, on the other end machine a conical tapered collar that slips over the rod O.D. This setup will put the rod on exact center - with a cleaning patch on the jag. Make a mounting block to position a second rod parallel to the first and abount 3/4 inch gap away from the first rod. Now the thing looks like a 2 foot long hairpin. At the "jag end" of the second rod drill & tap a small hole, at a right angle to the rod center line - like a #6 and thread a screw into the hole. Now, push the jag/patch into the bore, center the rod at the barrel end with the conical tapered collar, and measure from the barrel O.D. to the tip of the #6 screw. You can turn the screw in/out to change the gap between the screw tip and the barrel O.D. surface. Now rotate the tool around the barrel taking additional measurements. If the measurements are all the same - you are happy! If the measurement vary (much) the barrel bore is off center - you are sad!
The good news is that with this tool the barrel O.D. can be filed to bring the barrel bore to exact center. I hope this makes sense.
Jim